Average Error: 30.1 → 0.2
Time: 5.6s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}
double f(double x) {
        double r486748 = x;
        double r486749 = 1.0;
        double r486750 = r486748 + r486749;
        double r486751 = sqrt(r486750);
        double r486752 = sqrt(r486748);
        double r486753 = r486751 - r486752;
        return r486753;
}
↓
double f(double x) {
        double r486754 = 1.0;
        double r486755 = 0.0;
        double r486756 = r486754 + r486755;
        double r486757 = x;
        double r486758 = r486757 + r486754;
        double r486759 = sqrt(r486758);
        double r486760 = sqrt(r486757);
        double r486761 = r486759 + r486760;
        double r486762 = r486756 / r486761;
        double r486763 = sqrt(r486762);
        double r486764 = sqrt(r486754);
        double r486765 = r486763 * r486764;
        double r486766 = sqrt(r486761);
        double r486767 = r486765 / r486766;
        return r486767;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.1

    \[\sqrt{x + 1} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--29.9

    \[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\]

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.3

    \[\leadsto \color{blue}{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}}}\]
  7. Using strategy rm

  8. Applied sqrt-div0.3

    \[\leadsto \sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \color{blue}{\frac{\sqrt{1 + 0}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}\]

  9. Applied associate-*r/0.2

    \[\leadsto \color{blue}{\frac{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1 + 0}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}\]

  10. Simplified0.2

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\]

  11. Final simplification0.2

    \[\leadsto \frac{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))